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Omni-MATH / Let n to be a positive integer. Given a set {a₁,a₂,…,a_(n)} of integers, where a_(i)∈{0,1,2,3,…,2^(n)−1}, ∀i, we associate to each of its subsets the sum of its elements; particula…

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Let nn to be a positive integer. Given a set {a1,a2,…,an}\{ a_1, a_2, \ldots, a_n \} of integers, where ai∈{0,1,2,3,…,2n−1},a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \}, ∀i\forall i, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to 00. If all of these sums have different remainders when divided by 2n2^n, we say that {a1,a2,…,an}\{ a_1, a_2, \ldots, a_n \} is [i]nn-complete[/i]. For each nn, find the number of [i]nn-complete[/i] sets.
Plain-text mathematical notation (without MathML)
Let n to be a positive integer. Given a set {a₁,a₂,…,a_(n)} of integers, where a_(i)∈{0,1,2,3,…,2^(n)−1}, ∀i, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to 0. If all of these sums have different remainders when divided by 2^(n), we say that {a₁,a₂,…,a_(n)} is [i]n-complete[/i].

For each n, find the number of [i]n-complete[/i] sets.
Original LaTeX notation
Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different remainders when divided by $2^n$, we say that $\{ a_1, a_2, \ldots, a_n \} $ is [i]$n$-complete[/i].

For each $n$, find the number of [i]$n$-complete[/i] sets.

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task_id
76a3fda3-bdd8-5454-b01f-ce09b375fece
task_key
test--76a3fda3-bdd8-5454-b01f-ce09b375fece
task_revision_id
1
config / split
/ test

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